Article
| Title: | Solution of 3D contact shape optimization problems with Coulomb friction based on TFETI (English) |
| Author: | Markopoulos, Alexandros |
| Author: | Beremlijski, Petr |
| Author: | Vlach, Oldřich |
| Author: | Sadowská, Marie |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 68 |
| Issue: | 4 |
| Year: | 2023 |
| Pages: | 405-424 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | The present paper deals with the numerical solution of 3D shape optimization problems in frictional contact mechanics. Mathematical modelling of the Coulomb friction problem leads to an implicit variational inequality which can be written as a fixed point problem. Furthermore, it is known that the discretized problem is uniquely solvable for small coefficients of friction. Since the considered problem is nonsmooth, we exploit the generalized Mordukhovich's differential calculus to compute the needed subgradient information.\looseness -1 \endgraf The state problem is solved using successive approximations combined with the Total FETI (TFETI) method. The latter is based on tearing the bodies into ``floating'' subdomains, discretization by finite elements, and solving the resulting quadratic programming problem by augmented Lagrangians. \endgraf The presented numerical experiments demonstrate our method's power and the importance of the proper modelling of 3D frictional contact problems. The state problem solution and the sensitivity analysis process were implemented in parallel. (English) |
| Keyword: | shape optimization |
| Keyword: | nonsmooth optimization |
| Keyword: | contact problem |
| Keyword: | Coulomb's friction |
| Keyword: | TFETI method |
| MSC: | 65K05 |
| MSC: | 65K10 |
| MSC: | 74M10 |
| idZBL: | Zbl 07729504 |
| idMR: | MR4612740 |
| DOI: | 10.21136/AM.2022.0124-22 |
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| Date available: | 2023-07-10T14:10:23Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151702 |
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| Reference: | [1] Beremlijski, P., Haslinger, J., Kočvara, M., Kučera, R., Outrata, J. V.: Shape optimization in three-dimensional contact problems with Coulomb friction.SIAM J. Optim. 20 (2009), 416-444. Zbl 1186.49028, MR 2507130, 10.1137/080714427 |
| Reference: | [2] Beremlijski, P., Haslinger, J., Kočvara, M., Outrata, J.: Shape optimization in contact problems with Coulomb friction.SIAM J. Optim. 13 (2002), 561-587. Zbl 1025.49026, MR 1951035, 10.1137/S1052623401395061 |
| Reference: | [3] Beremlijski, P., Haslinger, J., Outrata, J. V., Pathó, R.: Shape optimization in contact problems with Coulomb friction and a solution-dependent friction coefficient.SIAM J. Control Optim. 52 (2014), 3371-3400. Zbl 1307.49040, MR 3272620, 10.1137/130948070 |
| Reference: | [4] Beremlijski, P., Markopoulos, A.: On solution of 3D contact shape optimization problems with Coulomb friction based on domain decomposition.EngOpt 2014 4th International Conference on Engineering Optimization IDMEC - Instituto de Engenharia Mecanica, Lisboa (2015), 465-470. 10.1201/b17488-82 |
| Reference: | [5] Clarke, F. H.: Optimization and Nonsmooth Analysis.Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, New York (1983). Zbl 0582.49001, MR 0709590, 10.1137/1.9781611971309 |
| Reference: | [6] Dostál, Z., Kozubek, T., Markopoulos, A., Brzobohatý, T., Vondrák, V., Horyl, P.: Theoretically supported scalable TFETI algorithm for the solution of multibody 3D contact problems with friction.Comput. Methods Appl. Mech. Eng. 205-208 (2012), 110-120. Zbl 1239.74064, MR 2872030, 10.1016/j.cma.2011.02.015 |
| Reference: | [7] Dostál, Z., Kozubek, T., Sadowská, M., Vondrák, V.: Scalable Algorithms for Contact Problems.Advances in Mechanics and Mathematics 36. Springer, New York (2016). Zbl 1383.74002, MR 3586594, 10.1007/978-1-4939-6834-3 |
| Reference: | [8] Farhat, C., Roux, F.-X.: An unconventional domain decomposition method for an efficient parallel solution of large-scale finite element systems.SIAM J. Sci. Stat. Comput. 13 (1992), 379-396. Zbl 0746.65086, MR 1145192, 10.1137/0913020 |
| Reference: | [9] Haslinger, J., Kozubek, T., Kučera, R., Peichl, G.: Projected Schur complement method for solving non-symmetric systems arising from a smooth fictitious domain approach.Numer. Linear Algebra Appl. 14 (2007), 713-739. Zbl 1199.65102, MR 2361187, 10.1002/nla.550 |
| Reference: | [10] Kučera, R.: Minimizing quadratic functions with separable quadratic constraints.Optim. Methods Softw. 22 (2007), 453-467. Zbl 1136.65062, MR 2319243, 10.1080/10556780600609246 |
| Reference: | [11] Kučera, R., Motyčková, K., Markopoulos, A., Haslinger, J.: On the inexact symmetrized globally convergent semi-smooth Newton method for 3D contact problems with Tresca friction: The R-linear convergence rate.Optim. Methods Softw. 35 (2020), 65-86. Zbl 07136209, MR 4032941, 10.1080/10556788.2018.1556659 |
| Reference: | [12] Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation. I. Basic Theory.Grundlehren der Mathematischen Wissenschaften 330. Springer, Berlin (2006),\99999DOI99999 10.1007/3-540-31247-1 . Zbl 1100.49002, MR 2191744 |
| Reference: | [13] Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation. II. Applications.Grundlehren der Mathematischen Wissenschaften 331. Springer, Berlin (2006). Zbl 1100.49002, MR 2191745, 10.1007/3-540-31246-3 |
| Reference: | [14] Mordukhovich, B. S.: Variational Analysis and Applications.Springer Monographs in Mathematics. Springer, Cham (2018). Zbl 1402.49003, MR 3823783, 10.1007/978-3-319-92775-6 |
| Reference: | [15] Myśliński, A.: Topology optimization of elasto-plastic contact problems.AIP Conf. Proc. 2239 (2020), Article ID 020031, 2 pages. 10.1063/5.0008122 |
| Reference: | [16] Outrata, J. V., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results.Nonconvex Optimization and Its Applications 28. Kluwer, Dordrecht (1998). Zbl 0947.90093, MR 1641213, 10.1007/978-1-4757-2825-5 |
| Reference: | [17] Říha, L., Brzobohatý, T., Markopoulos, A.: Hybrid parallelization of the total FETI solver.Adv. Eng. Softw. 103 (2017), 29-37. 10.1016/j.advengsoft.2016.04.004 |
| Reference: | [18] Rockafellar, R. T., Wets, R. J.-B.: Variational Analysis.Grundlehren der Mathematischen Wissenschaften 317. Springer, Berlin (1998). Zbl 0888.49001, MR 1491362, 10.1007/978-3-642-02431-3 |
| Reference: | [19] Schramm, H., Zowe, J.: A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results.SIAM J. Optim. 2 (1992), 121-152. Zbl 0761.90090, MR 1147886, 10.1137/0802008 |
| Reference: | [20] Sharma, A., Rangarajan, R.: A shape optimization approach for simulating contact of elastic membranes with rigid obstacles.Int. J. Numer. Methods Eng. 117 (2019), 371-404. MR 3903330, 10.1002/nme.5960 |
| Reference: | [21] Vondrák, V., Kozubek, T., Markopoulos, A., Dostál, Z.: Parallel solution of contact shape optimization problems based on total FETI domain decomposition method.Struct. Multidiscip. Optim. 42 (2010), 955-964. Zbl 1274.74407, MR 2735250, 10.1007/s00158-010-0537-3 |
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